Grid Approximation of Singularly Perturbed Elliptic Convection-Diffusion Equations in Unbounded Domains

In the quarter plane $\{(x,y): x, y \ge 0 \}$, we consider the Dirichlet problem for a singularly perturbed elliptic convection-diffusion equation. The highest derivatives of the equation and the first derivative along the $y$-axis contain respectively the parameters $\eps_1$ and $\eps_2$, which take arbitrary values from the half-open interval $(0,1]$ and the segment $[-1,1]$. For small values of the parameter $\eps_1$, a boundary layer appears in a neighbourhood of the domain boundary. Depending on the ratio between the parameters $\eps_1$ and $\eps_2$, this layer may be regular, parabolic or hyperbolic. Besides the boundary-layer scale controlled by the perturbation parameters, one can observe a resolution scale, which is specified by the "width" of the domain on which the problem is to be solved on a computer. It turns out that, for solutions of the boundary value problem and of a formal difference scheme (i.e., a scheme on meshes with an infinite number of nodes) considered on the bounded subdomains of interest (referred to as the resolution subdomains), the domains of essential dependence, i.e., such domains outside which the finite variation of the solution causes relatively small disturbances of the solution on the resolution subdomains, are bounded uniformly with respect to the vector-parameter $\overline{\eps}=(\eps_1, \eps_2$). Using the conception of region of essential solution dependence'', we design a constructive finite difference scheme (i.e., a scheme on meshes with a finite number of nodes) that converges $\overline{\eps}$-uniformly on the bounded resolution subdomains.

Acknowledgements This work has been supported by the Russian Foundation for Basic Research under grant No. 04-01-00578.